Second Interlude
Modelling the Syllogism
Second InterludeModelling the Syllogism
People do not understand how that
which is at variance with itself, agrees with itself.
Heraclitus
It is impossible for the same attribute at once to belong and
not to belong to the same thing and in the same relation.
Aristotle
I
n this interlude, I show how relational logic can be used to model syllogistic
reasoning, originally developed by Aristotle some 2,300 years ago. Aristotle
was concerned with various properties of things and with what conclusions
can be made about these properties from assumed premises. These properties
can quite easily be represented in relational logic.
However, I am not here primarily concerned with the nature of the properties
of the entities under discussion. I wish to model the underlying structure of
Aristotle’s thought processes. This can best be done by studying the concepts of
the syllogism and their relationships to each other.
Definitions
A syllogism consists of three propositions (also called statements or
sentences), called the major premise, minor premise, and conclusion,
respectively. Each proposition has two terms called the subject and predicate.
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The terms in a proposition are related to each other in four different ways,
shown in Table 34.
Class
Attribute names
Syllogistic propositions
Name Form
A
All S are P
Diagram
P
S
E
All S are not P
P
S
Attribute values
I
O
Some S are P
S
S
P
or
P
S
P
or
P
Some S are not P
S
Table 34. Types of propositions in syllogisms
Each of these propositions has a number of dualistic attributes that
characterize the propositions. They are grouped together in pairs depending on
whether A is paired with E, I, or O. Table 35, which is an extension of Table 34,
shows these attributes. Any two of these attributes uniquely defines the
proposition. So we could call them defining attributes, with the third being
derived from the other two.
Class
Attribute names
Attribute values
Syllogistic propositions
Name Universality
A
universal
E
universal
I
particular
O
particular
Positivity
positive
negative
positive
negative
Symmetry
asymmetrical
symmetrical
symmetrical
asymmetrical
Table 35. Attributes of propositions
Aristotle called the symmetrical propositions convertible because they are
equivalent when the terms are interchanged. A and E are also convertible into
weaker forms, I and O, respectively. Furthermore, if we assume Aristototle’s rules
of logic, A and O and E and I are contradictory; they exclude each other.
One other property of these propositions relates to the terms in the
proposition, rather than the propositions themselves. A term is distributed if, in
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some sense, it refers to all entities with the particular property (called a class),
otherwise it is undistributed. The subject of universal propositions and the
predicate of negative propositions are distributed. How these definitions relate
to syllogistic propositions is given in Table 36, attributes that can be seen quite
clearly from the diagrams in Table 34.
Class
Attribute names
Attribute values
Syllogistic propositions
Name Subject
A
distributed
E
distributed
I
undistributed
O
undistributed
Predicate
undistributed
distributed
undistributed
distributed
Table 36. Distribution of terms in syllogism
The terms of the three propositions of the syllogism are related to each other
in two ways:
1. One term is common to the major and minor premises; it is called the
middle term (M).
2. The predicate (P) of the conclusion is the major term of the syllogism and
the subject (S) is the minor term, because they are the nonmiddle terms in
the major and minor premises, respectively.
As each proposition has one of four types and as there are three propositions
in each syllogism, there are 43 = 64 different syllogistic forms, called moods.
These are naturally called AAA, AAE, AAI, etc.
In addition, the syllogism can have one of four figures, depending on whether
the middle term is the subject or predicate in the major and minor premises.
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(Curiously, for some reason, Aristotle only recognized three of these figures; the
fourth was discovered only in the Middle Ages.)
Class
Attribute names
Attribute values
Syllogistic figures
Name
Figure
I
M P
S M
S P
II
P M
S M
S P
III
M P
M S
S P
IV
P M
M S
S P
Table 37. Syllogistic figures
There are thus 64 x 4 = 256 possible syllogisms in total.
Aristotle examined each mood and figure in turn to determine whether it was
valid or not. He then derived a number of common properties of these
syllogisms, which can be called rules of deductions. I reverse this process here.
These are the rules that Aristotle discovered:
1. Relating to premises irrespective of conclusion or figure
(a) No
inference can be made from two particular premises.
(b) No
inference can be made from two negative premises.
2. Relating to propositions irrespective of figure
(a) If
one premise is particular, the conclusion must be particular.
(b) If
one premise is negative, the conclusion must be negative.
3. Relating to the distribution of terms
(a) The
middle term must be distributed at least once.
(b) A
predicate distributed in the conclusion must be distributed in
the major premise.
(c) A
subject distributed in the conclusion must be distributed in the
minor premise.
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Application of rules
We can now apply these rules of inference to determine the validity of each
mood in each figure.
As the first rule applies irrespective of conclusion or figure, we need consider
just sixteen pairs of premises.
By rule 1(a), these pairs of premises are invalid:
II
IO
OI
OO
By rule 1(b), these pairs of premises are invalid:
EE
EO
OE
OO
I could now show the sixteen premises in the form of a relation, indicating
which has an attribute value invalid by one or both of the two rules. However, it
is simpler to indicate invalidity by type style, bold and italics indicating
invalidity by rules 1(a) and 1(b), respectively. The nine potentially valid premises
are marked in plain style:
AA
EA
IA
OA
AE
EE
IE
OE
AI
EI
II
OI
AO
EO
IO
OO
Rule 2 applies to the moods of syllogisms. As there are four moods and nine
potentially valid pairs of premises from rule 1, we need to consider 36 moods.
By rule 2(a), these moods are invalid:
xIA
IxA
xOA
OxA
xIE
IxE
xOE
OxE
By rule 2(b), these pairs of premises are invalid:
xEA
ExA
xOA
OxA
xEI
ExI
xOI
OxI
Rule 2 therefore leaves 16 potentially valid moods of syllogism, the invalid
ones being marked in italics and bold, as for rule 1:
AAA
AIA
EAA
EIA
IAA
OAA
AAE
AIE
EAE
EIE
IAE
OAE
AAI
AII
EAI
EII
IAI
OAI
AAO
AIO
EAO
EIO
IAO
OAO
AEA
AOA
AEE
AOE
AEI
AOI
AEO
AOO
IEA
IEE
IEI
IEO
The third rule is rather more complicated as it concerns the distribution of
each of the three types of term in the syllogism for each mood and figure.
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First considering rule 3(a) about the distribution of the middle term. The
right-hand four columns in the following table show the distribution of the
middle term in the two premises. The ones that are invalid by this rule are the
ones that are undistibuted in both premises, indicated by italics in the table. The
other four columns show the sixteen potentially valid moods that we are
considering after eliminating the others by applying rules 1 and 2.
AAx
AEx
AIx
AOx
EAx
EEx
EIx
EOx
IAx
IEx
IIx
IOx
OAx
OEx
OIx
OOx
308
xxA
AAA
xxE
AAE
AEE
xxI
AAI
AII
EAE
IAI
xxO
AAO
AEO
AIO
AOO
EAO
I
D–U
D–D
D–U
D–D
D–U
II
U–U
U–D
U–U
U–D
D–U
III
D–D
D–D
D–U
D–U
D–D
IV
U–D
U–D
U–U
U–U
D–D
EIO
D–U
D–U
D–U
D–U
IAO
IEO
U–U
U–D
U–U
U–D
U–D
U–D
U–D
U–D
OAO
U–U
D–U
U–D
D–D
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Modelling the Syllogism
Rule 3(b) concerns the distribution of the predicate. The ones that are invalid
by this rule are the ones that are distributed in the conclusion, but not
distributed in the major premise, indicated by italics in the table.
AxA
AxE
AxI
AxO
ExA
ExE
ExI
ExO
IxA
IxE
IxI
IxO
OxA
OxE
OxI
OxO
xAx
AAA
AAE
AAI
AAO
xEx
xIx
xOx
AII
AIO
AOO
AEE
AEO
EAE
EAO
IAI
IAO
EIO
IEO
OAO
I
U–U
U–D
U–U
U–D
II
D–U
D–D
D–U
D–D
III
U–U
U–D
U–U
U–D
IV
D–U
D–D
D–U
D–D
D–D
D–D
D–D
D–D
D–D
D–D
D–D
D–D
U–U
U–D
U–U
U–D
U–U
U–D
U–U
U–D
D–D
U–D
D–D
U–D
Rule 3(c) is similar to 3(b) except that it applies to the subject in the minor
premise. The pairs of minor premise–conclusion that are invalid are again
indicated by italics in the table.
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xAA
xAE
xAI
xAO
xEA
xEE
xEI
xEO
xIA
xIE
xII
xIO
xOA
xOE
xOI
xOO
Axx
AAA
AAE
AAI
AAO
Exx
Ixx
Oxx
IAI
IAO
OAO
EAE
EAO
AEE
AEO
AII
AIO
IEO
EIO
AOO
I
D–D
D–D
D–U
D–U
II
D–D
D–D
D–U
D–U
III
U–D
U–D
U–U
U–U
IV
U–D
U–D
U–U
U–U
D–D
D–D
D–D
D–D
D–U
D–U
D–U
D–U
U–U
U–U
U–U
U–U
U–U
U–U
U–U
U–U
U–U
U–U
D–U
D–U
Eliminating the syllogisms, both mood and figure, made invalid by rule 3
leaves just 12 moods and 25 syllogisms. One of them, AAO in figure IV, is in
rather a curious position. Aristotle’s rules allow it. However, it is only
conditionally valid. As this is a syllogism of figure IV, and as Aristotle did not
discover this figure, it is not surprising that he did not consider it.
AAO in figure IV is only valid if P is an actual subset of M and if M is an
actual subset of S. However, if P, M, and S are equivalent, there are no S that are
not P. So we need to eliminate it from the list for this reason, not covered by
Aristotle’s rules. It is only conditionally valid.
There are five syllogisms that have a universal conclusion. So there are also
five corresponding syllogisms with a particular conclusion, which we can also
eliminate, as they are weak forms. These are:
Strong
AAA I
EAE I
AEE II
EAE II
AEE IV
Weak
AAI
I
EAO I
AEO II
EAO II
AEO IV
This leaves us with 20 valid syllogisms, found by Aristotle and his successors:
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Modelling the Syllogism
First figure
AAA, EAE, AII, EIO
Second figure
EAE, AEE, EIO, AOO
Third figure
AAI, IAI, AII, EAO, OAO, EIO
Fourth figure
AAI, AEE, IAI, EAO, EIO
Students in the Middle Ages were expected to know all these by heart. For
instance, the statutes of the University of Oxford in the fourteenth century
included this rule: “Batchelors and Masters of Arts who do not follow Aristotle’s
philosophy are subject to a fine of 5s. for each point of divergence, as well as for
infractions of the rules of the Organum”.1
Not surprising therefore that they needed a mnemonic to remember this
rather arbitrary set of letters:
Barbara, Celarent, Darii, Ferioque, prioris:
Cesare, Camestres, Festino, Baroko, secundae:
Tertia, Darapti, Disamis, Datisi, Felpaton, Bokardo, Ferison, habet:
Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison.2
I don’t know what Tertia (EIA in the third figure) is doing here. She was
eliminated by rule 2(a). Curiously, C. W. Kilmister, whose book Language, Logic
and Mathematics this mnemonic is taken from, did not point out the error.
These syllogisms can be further reduced because propositions E and I are
symmetrical; the terms in these propositions can be interchanged. When one or
both premises is symmetrical, the mood stays unchanged, but the figure changes.
When the conclusion is symmetrical, interchanging the subject and predicate
means that the major and minor premises must change position, resulting in a
change in both mood and figure.
This means that there are just eight core syllogisms out of the 256 candidates
that we started with.
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Modelling the Syllogism
Syllogism: Mood (figure)
AAA (I)
AII (I)
EAE (I)
EIO (I)
AOO (II)
AAI (III)
EAO (III)
OAO (III)
Equivalent to
AAI (IV) ≡ AAI (I) [weak form]
AII (III), IAI (III), IAI (IV)
EAE (II), AEE (II), AEE (IV)
EIO (II), EIO (IIII), EIO (IV)
—
Itself
EAO (IV)
—
To sum up, the table on the next page shows all 256 syllogisms, which are
valid, which are equivalent to a valid syllogism, and which rule first eliminated
them from the list.
However, all this is rather abstract and mechanical. It tells us little about the
meanings of this inferences. So tables 38 on page 314 and 39 on page 320
provide a list of examples of each of the eight valid syllogisms, together with a
corresponding Euler-Venn diagram. Note that because I and O can be
represented in two ways in Euler-Venn diagrams, as illustrated in Table 34 on
page 304, there are sometimes four or five corresponding diagrams for one
particular mood and figure.
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Modelling the Syllogism
Class of syllogisms
Figure
Mood
AAA
AAE
AAI
AAO
AEA
AEE
AEI
AEO
AIA
AIE
AII
AIO
AOA
AOE
AOI
AOO
EAA
EAE
EAI
EAO
EEA
EEE
EEI
EEO
EIA
EIE
EII
EIO
EOA
EOE
EOI
EOO
I
Valid
3(b)
Weak
3(b)
3(b)
3(b)
Valid
3(b)
3(b)
Valid
Weak
II
III
3(c)
3(b,c)
3(a)
Valid
3(b)
2(b)
Equiv 3(b)
2(b)
Weak 3(b)
2(a)
2(a)
Equiv
3(a)
3(b)
2(a,b)
2(a)
2(b)
Valid
3(b)
2(b)
Equiv 3(c)
2(b)
Weak Valid
1(b)
Valid
2(a,b)
2(a)
2(b)
Equiv Equiv
1(b)
Draft 14 February 2004
Figure
IV
3(c)
3(c)
Equiv
Cond
Mood
IAA
IAE
IAI
IAO
IEA
Equiv IEE
IEI
Weak IEO
IIA
IIE
III
3(a)
IIO
IOA
IOE
IOI
3(a) IOO
OAA
3(c) OAE
OAI
Equiv OAO
OEA
OEE
OEI
OEO
OIA
OIE
OII
Equiv OIO
OOA
OOE
OOI
OOO
I
3(a)
3(a,b)
3(b)
II
III
IV
2(a)
2(a)
3(a) Equiv Equiv
3(a,b) 3(b) 3(b)
2(a,b)
2(a)
2(b)
3(b)
3(b) 3(b)
1(a)
1(a)
3(a)
2(a,b)
2(a)
2(b)
3(b)
Valid
3(b)
1(b)
1(a)
1(a,b)
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314
All M are P
Some S are M
Some S are P
[Some S are not P]
All M are P
Some S are M
Some S are P
[All S are P]
AII (I)
AII (I)
All M are not P
All S are M
All S are not P
EAE (I)
Table 38. Valid syllogisms sorted by mood and figure
All Catholics are Christians
Some Germans are Catholics
Some Germans are Christians
[Some Germans are not Christians]
All mathematicians are human
Some meditators are mathematicians
Some meditators are human
[All meditators are human]
All primates are not bears
All humans are primates
All humans are not bears
Class of valid syllogisms (Sheet 1 of 6)
Mood (Figure) Form
Example
AAA (I)
All M are P
All primates are mammals
All S are M
All humans are primates
All S are P
All humans are mammals
S
P
S
S
M
M
S
Diagram
M
M
P
P
P
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Modelling the Syllogism
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All M are not P
Some S are M
Some S are not P
[Some S are P]
All M are not P
Some S are M
Some S are not P
[All P are S]
EIO (I)
EIO (I)
All M are P
Some S are M
Some S are P
[All M are S]
[All P are S]
AII (I)
Table 38. Valid syllogisms sorted by mood and figure
All men are not women
Some mathematicians are men
Some mathematicians are not women
[Some mathematicians are women]
All meat eaters are not elephants
Some wild animals are meat eaters
Some wild animals are not elephants
[All elephants are wild animals]
All humans are primates
Some mammals are humans
Some humans are primates
[All humans are mammals]
[All primates are mammals]
Class of valid syllogisms (Sheet 2 of 6)
Mood (Figure) Form
Example
AII (I)
All M are P
All humans are mammals
Some S are M
Some mammals are primates
Some S are P
Some primates are humans
[All P are S]
[All humans are primates]
M
M
M
M
Diagram
S
P
P
S
P
S
S
P
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Modelling the Syllogism
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316
Table 38. Valid syllogisms sorted by mood and figure
Class of valid syllogisms (Sheet 3 of 6)
Mood (Figure) Form
Example
EIO (I)
All M are not P
All bears are not meditators
Some S are M
Some mathematicians are meditators
Some S are not P
Some mathematicians are not bears
[All S are not P]
[All mathematicians are not bears]
EIO (I)
All M are not P
All elephants are not meat eaters
Some S are M
Some wild animals are elephants
Some S are not P
Some wild animals are not meat eaters
[All M are S]
[All elephants are wild animals]
All humans are not bears
EIO (I)
All M are not P
Some primates are humans
Some S are M
Some primates are not bears
Some S are not P
[All primates are not bears]
[All S are not P]
AOO (II)
All P are M
All Catholics are Christians
Some S are not M
Some Germans are not Christians
Some S are not P
Some Germans are not Catholics
[Some S are P]
[Some Germans are Catholics]
All tigers are meat eaters
AOO (II)
All P are M
Some wild animals are not meat eaters
Some S are not M
Some wild animals are not tigers
Some S are not P
[All tigers are wild animals]
[All P are S]
S
S
S
S
M
M
M
P
Diagram
M
P
P
S
M
P
P
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All tigers are wild animals
All tigers are meat eaters
Some meat eaters are wild animals
[Some wild animals are not meat eaters]
[Some meat eaters are not wild animals]
All humans are primates
All humans are mammals
Some mammals are primates
[All primates are mammals]
All humans are primates
Some mammals are primates
Some mammals are human
[All primates are mammals]
[All humans are mammals]
Table 38. Valid syllogisms sorted by mood and figure
All M are P
All M are S
Some S are P
[Some P are not S]
[Some S are not P]
All M are P
All M are S
Some S are P
[All P are S]
AAI (III)
AAI (III)
All P are M
Some S are not M
Some S are not P
[All M are S]
[All P are S]
AOO (II)
Class of valid syllogisms (Sheet 4 of 6)
Mood (Figure) Form
Example
AOO (II)
All P are M
All elephants are wild animals
Some S are not M
Some meat eaters are not wild animals
Some S are not P
Some meat eaters are not elephants
[All S are not P]
[All elephants are not meat eaters]
S
S
M
M
P
Diagram
P
M
S
P
S
P
M
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Modelling the Syllogism
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Table 38. Valid syllogisms sorted by mood and figure
Class of valid syllogisms (Sheet 5 of 6)
Mood (Figure) Form
Example
EAO (III)
All M are not P
All elephants are not meat eaters
All M are S
All elephants are wild animals
Some S are not P
Some wild animals are not meat eaters
[Some S are P]
[Some wild animals are meat eaters]
EAO (III)
All M are not P
All men are not women
All M are S
All men are human
Some S are not P
Some humans are not women
[All P are S]
[All women are human]
All primates are not bears
EAO (III)
All M are not P
All primates are humans
All M are S
Some humans are not bears
Some S are not P
[All humans are not bears]
[All S are not P]
OAO (III)
Some M are not P
Some Germans are not Christians
All M are S
All Catholics are Christians
Some S are not P
Some Christians are not Catholics
[Some P are not S]
[Some Christians are Catholics]
Some meditators are not mathematicians
OAO (III)
Some M are not P
All meditators are human
All M are S
Some humans are not mathematicians
Some S are not P
[All mathematicians are humans]
[Some P are not M]
S
M
M
S
S
M
M
S
M
P
S
P
Diagram
P
P
P
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Modelling the Syllogism
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Table 38. Valid syllogisms sorted by mood and figure
Class of valid syllogisms (Sheet 6 of 6)
Mood (Figure) Form
Example
OAO (III)
Some M are not P
Some primates are not humans
All M are S
All primates are mammals
Some S are not P
Some mammals are not humans
[All P are M]
[All humans are primates]
[All P are S]
[All humans are mammals]
P
Diagram
M
S
Second Interlude
Modelling the Syllogism
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320
P
M
M
M
S
P
S
P
S
All M are P
Some S are M
Some S are P
[All M are S]
[All P are S]
All P are M
Some S are not M
Some S are not P
[All M are S]
[All P are S]
All humans are primates
Some mammals are humans
Some humans are primates
[All humans are mammals]
[All primates are mammals]
All humans are primates
Some mammals are primates
Some mammals are human
[All primates are mammals]
[All humans are mammals]
Table 39. Valid syllogisms sorted by type of Euler-Venn diagram
All M are P
Some S are M
Some S are P
[All P are S]
Form
All M are P
All S are M
All S are P
All humans are mammals
Some mammals are primates
Some primates are humans
[All humans are primates]
Class of valid syllogisms (Sheet 1 of 5)
Diagram
Example
All primates are mammals
P
All humans are primates
M
All humans are mammals
S
AOO (II)
AII (I)
AII (I)
Mood (Figure)
AAA (I)
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Modelling the Syllogism
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S
M
M
S
S
M
P
M
S
P
P
P
All M are not P
All S are M
All S are not P
All M are not P
Some S are M
Some S are not P
[All S are not P]
All M are not P
All M are S
Some S are not P
[All S are not P]
All primates are not bears
All humans are primates
All humans are not bears
All humans are not bears
Some primates are humans
Some primates are not bears
[All primates are not bears]
All primates are not bears
All primates are humans
Some humans are not bears
[All humans are not bears]
Table 39. Valid syllogisms sorted by type of Euler-Venn diagram
OAO (III)
Some M are not P
All M are S
Some S are not P
[All P are M]
[All P are S]
Some primates are not humans
All primates are mammals
Some mammals are not humans
[All humans are primates]
[All humans are mammals]
EAO (III)
EIO (I)
EAE (I)
Mood (Figure)
AAI (III)
Form
All M are P
All M are S
Some S are P
[All P are S]
Class of valid syllogisms (Sheet 2 of 5)
Diagram
Example
All humans are primates
S
All humans are mammals
P
Some mammals are primates
M
[All primates are mammals]
Second Interlude
Modelling the Syllogism
321
322
Form
All M are P
Some S are M
Some S are P
[Some S are not P]
All P are M
Some S are not M
Some S are not P
[Some S are P]
Some M are not P
All M are S
Some S are not P
[Some P are not S]
All M are P
Some S are M
Some S are P
[All S are P]
Some M are not P
All M are S
Some S are not P
[Some P are not M]
Table 39. Valid syllogisms sorted by type of Euler-Venn diagram
Class of valid syllogisms (Sheet 3 of 5)
Diagram
Example
All Catholics are Christians
Some Germans are Catholics
S
M P
Some Germans are Christians
[Some Germans are not Christians]
All Catholics are Christians
Some Germans are not Christians
S
P M
Some Germans are not Catholics
[Some Germans are Catholics]
Some Germans are not Christians
All Catholics are Christians
S M
P
Some Christians are not Catholics
[Some Christians are Catholics]
All mathematicians are human
P
Some meditators are mathematicians
S
M
Some meditators are human
[All meditators are human]
Some meditators are not mathematicians
S
All meditators are human
M
P
Some humans are not mathematicians
[All mathematicians are humans]
OAO (III)
AII (I)
OAO (III)
AOO (II)
Mood (Figure)
AII (I)
Second Interlude
Modelling the Syllogism
14 February 2004
14 February 2004
Form
All M are not P
Some S are M
Some S are not P
[Some S are P]
All M are not P
Some S are M
Some S are not P
[All P are S]
All M are not P
Some S are M
Some S are not P
[All M are S]
All P are M
Some S are not M
Some S are not P
[All S are not P]
All M are not P
All M are S
Some S are not P
[Some S are P]
Table 39. Valid syllogisms sorted by type of Euler-Venn diagram
Class of valid syllogisms (Sheet 4 of 5)
Diagram
Example
All men are not women
Some mathematicians are men
M
S
P
Some mathematicians are not women
[Some mathematicians are women]
All meat eaters are not elephants
S
Some wild animals are meat eaters
M
P
Some wild animals are not elephants
[All elephants are wild animals]
All elephants are not meat eaters
S
Some wild animals are elephants
M
P
Some wild animals are not meat eaters
[All elephants are wild animals]
All elephants are wild animals
M
Some meat eaters are not wild animals
S
P
Some meat eaters are not elephants
[All elephants are not meat eaters]
All elephants are not meat eaters
S
All elephants are wild animals
M
P
Some wild animals are not meat eaters
[Some wild animals are meat eaters]
EAO (III)
AOO (II
EIO (I)
EIO (I)
Mood (Figure)
EIO (I)
Second Interlude
Modelling the Syllogism
323
324
Form
All M are not P
Some S are M
Some S are not P
[All S are not P]
All P are M
Some S are not M
Some S are not P
[All P are S]
All M are P
All M are S
Some S are P
[Some P are not S]
[Some S are not P]
All M are not P
All M are S
Some S are not P
[All P are S]
Table 39. Valid syllogisms sorted by type of Euler-Venn diagram
Class of valid syllogisms (Sheet 5 of 5)
Diagram
Example
All bears are not meditators
Some mathematicians are meditators
M
S
P
Some mathematicians are not bears
[All mathematicians are not bears]
All tigers are meat eaters
Some wild animals are not meat eaters
S
P
M
Some wild animals are not tigers
[All tigers are wild animals]
All tigers are wild animals
All tigers are meat eaters
S
M
P
Some meat eaters are wild animals
[Some wild animals are not meat eaters]
[Some meat eaters are not wild animals]
All men are not women
S
All men are human
M
P
Some humans are not women
[All women are human]
EAO (III)
AAI (III)
AOO (II)
Mood (Figure)
EIO (I)
Second Interlude
Modelling the Syllogism
14 February 2004